Abstract Curvature Invariants and the Cartan Equivalence Problem

Curvature invariants remove coordinate labels, but not every invariant package has the same separating power. Unequal corresponding scalar polynomial invariants prove that two metrics are not locally isometric; equal scalar polynomial invariants need not prove that they are. In Lorentzian geometry, a nonflat plane wave can have vanishing scalar invariants (VSI), just like flat spacetime. The example below shows why. Cartan invariants retain more information by comparing curvature and its derivatives in carefully normalized frames.

Required background. Affine Connections and Covariant Derivatives supplies covariant differentiation of tensors; Parallel Transport Along Curves supplies parallel fields and transported frames along a chosen curve.

Helpful background. Calculus and Linear Algebra for Relativity supplies linear maps, eigenstructure, and index notation; the invariant contractions and canonical-frame procedure are developed here. Exterior Algebra and Differential Forms supplies alternating forms and wedge products used in frame-based curvature notation.

Scalar polynomial invariants

A scalar polynomial curvature invariant is a complete contraction of a polynomial in the Riemann tensor and its covariant derivatives. Examples include

$$R, \qquad R_{ab}R^{ab}, \qquad R_{abcd}R^{abcd}, \qquad \nabla_eR_{abcd}\nabla^eR^{abcd}.$$

Each is unchanged under a coordinate transformation. Consequently, if corresponding invariants or their functional relations disagree, the metrics cannot be locally isometric. The converse is the delicate direction. Contraction discards the orientation of curvature operators relative to null directions, and an indefinite metric permits nonzero tensors whose every scalar contraction vanishes.

Eigenvalues and algebraic types of curvature operators retain more information than a short scalar list, but equality of those data is still only a necessary test. A sharper four-dimensional result concerns $\mathcal I$-nondegeneracy: the absence of a continuous family of inequivalent metrics with the same complete scalar-invariant data. On regular neighborhoods where the algebraic types of curvature and the required derivatives remain constant, every Lorentzian metric is either $\mathcal I$-nondegenerate or belongs to the Kundt class, which admits a null geodesic congruence with zero expansion, shear, and twist. This is the scope of Coley, Hervik, and Pelavas 2009, Definition 1.2 and Theorems 2.1–2.3, pp. 2–4, PDF. Discrete ambiguities can remain even in the nondegenerate case; the theorem is not an unrestricted assertion that equality of scalar invariants proves an isometry.

A nonflat VSI pp-wave

On $\mathbb R^4$ with coordinates $(u,v,x,y)$, consider the smooth metric

$$ds^2=H(u,x,y)\,du^2+2\,du\,dv+dx^2+dy^2.$$

This has Lorentzian signature, with one negative and three positive eigenvalues after diagonalizing the $u,v$ block. Its only potentially nonzero independent curvature components are aligned with the null covector $du$, whose raised vector is $\partial_v$; in a convenient coordinate frame,

$$R_{uiuj}=-\frac12\partial_i\partial_jH, \qquad i,j\in\{x,y\}.$$

Choose $H=(x^2-y^2)/a^2$ with a constant length $a\gt0$ and coordinates of length dimension. Then $R_{uxux}=-a^{-2}$ and $R_{uyuy}=+a^{-2}$, so the metric is not flat. To see exactly why scalar contractions fail, invert the $u,v$ block:

$$g^{uu}=0,\qquad g^{uv}=g^{vu}=1,\qquad g^{vv}=-H.$$

Every nonzero curvature component has two lower $u$ indices and no lower $v$ index. In a complete contraction, a lower $u$ must pair with a lower $v$ through $g^{uv}$, since $g^{uu}=g^{ui}=0$. The required curvature component with a $v$ slot is absent. Thus every positive-degree scalar polynomial made from Riemann alone vanishes.

The definition also includes covariant derivatives of curvature, so that argument alone is not enough. For this constant quadratic profile the only nonzero Christoffel symbols, up to symmetry, are

$$\Gamma^v{}_{ui}=\frac12\partial_iH, \qquad \Gamma^i{}_{uu}=-\frac12\partial_iH.$$

Substitution into the covariant derivative gives $\nabla R=0$: the transverse Hessian is constant, terms inserting a $v$ slot vanish, and the remaining terms cancel by curvature antisymmetry. All higher covariant derivatives therefore vanish too. This proves the full VSI claim for the example, including derivative invariants. Minkowski space, obtained with $H=0$, has the same vanishing scalar set but zero Riemann tensor. The general pp-wave comparison is discussed in Mergulhão and Batista 2020, §I, eq. (3), p. 2, PDF.

The example also separates a coordinate effect from an invariant failure. Riemann itself is nonzero, so no coordinate change can turn this pp-wave into Minkowski space. What fails is not coordinate invariance but completeness of scalar contraction as a classifier.

Cartan invariants and equivalence

Cartan’s method keeps components rather than contracting them all. At a regular point, one:

1. computes $R$ in a frame and uses the frame group to put it into canonical form;
2. records the residual isotropy subgroup;
3. computes $\nabla R,\nabla^2R,\ldots$, canonically normalizing the frame further;
4. stops when neither the number of functionally independent normalized components nor the residual isotropy group changes.

The resulting normalized frame components are Cartan invariants. For a pp-wave, frame-resolved curvature reveals the transverse Hessian of $H$ even though every scalar polynomial contraction vanishes. The displayed coordinate components $R_{uiuj}$ establish nonflatness; their numerical values are not themselves Cartan invariants until the frame has been normalized. Comparing normalized data, together with their functional dependence, gives a finite local-equivalence procedure in fixed dimension on regular neighborhoods where the relevant ranks and isotropy groups stabilize.

There are important qualifications. Frame components before normalization are not invariants. Discrete frame choices and isotropy must be tracked, not just the dimension of the remaining group. The method establishes local equivalence on suitable regular neighborhoods; global topology, boundaries, identifications, and completeness require separate analysis. The sequence of normalized components, functional relations, and residual groups is explained in Mergulhão and Batista 2020, §II, pp. 3–5, PDF.

A practical hierarchy of tests

Use inexpensive invariants first. A mismatch in $R$ or the Kretschmann scalar already rules out equivalence. If they agree, compare a wider scalar basis and the eigenvalue or alignment structure of curvature operators. When degeneracy remains possible—especially in null-aligned Kundt geometry—use normalized curvature components and covariant derivatives. A coordinate transformation need not be guessed at the outset.

This page explains the abstract discriminants. It does not classify individual black-hole metrics; that task belongs to the Spacetimes volume, where domains, parameters, and global identifications can be included.

Exercises

Explain why $R_{uxux}\ne0$ for the pp-wave with $H=(x^2-y^2)/a^2$ is enough to distinguish it from Minkowski space even though all scalar polynomial invariants vanish. Is the example Ricci flat?

Solution

The vanishing or nonvanishing of the full Riemann tensor is coordinate independent. Here $R_{uxux}=-\tfrac12\partial_x^2H=-a^{-2}$, so Riemann is nonzero at every point. Minkowski space has $R_{abcd}=0$. Therefore no local isometry exists. The only potentially nonzero Ricci component is $R_{uu}=-\tfrac12(H_{,xx}+H_{,yy})=0$, so the example is Ricci flat. It has nonzero Weyl curvature. Scalar contractions fail because the nonzero components are aligned with a null direction and disappear upon complete contraction, while all curvature derivatives vanish for this profile.

References

• Alan Coley, Sigbjørn Hervik, and Nicos Pelavas, “Spacetimes Characterized by Their Scalar Curvature Invariants,” Classical and Quantum Gravity 26, 025013, 2009. DOI.
• Thiago M. Mergulhão and Carlos Batista, “On the Equivalence of Spacetimes, the Cartan–Karlhede Algorithm,” Revista Brasileira de Ensino de Física 42, e20200041 (2020). DOI. Open PDF.